avo inversion in v x;z media - stanford university

Stanford Exploration Project, Report 97, July 8, 1998, pages 275–294

AVO inversion in V (x, z) media

Yalei Sun and Wenjie Dong1

keywords: 2.5-D Kirchhoff integral, AVO inversion, fluid-line section

ABSTRACT

We implement a new Kirchhoff-typed AVO inversion scheme in V (x, z) media.The WKBJ Green’s function is calculated using a finite-difference scheme. Wepropose a pair of Kirchhoff inversion operators which have more obvious physicalmeaning. By analyzing the Kirchhoff inversion operator, we find out an uniquerelationship between the weighting function and the kinematic equation, whichis very important to recover the true amplitude of the reflection coefficient. Ourscheme is a two-step AVO inversion approach. Common-image gathers (CIG)are generated in the first step. These common-image gathers can be used toupdate the velocity model and reduce the influence of velocity error in the finalAVO inversion results. AVO intercepts and slopes are estimated in the secondstep using a least-squares procedure. Finally, a fluid-line section is generated tohighlight the existence of Vp/Vs anomaly. One dipping-layered synthetic exampledemonstrates the accuracy of our scheme and the influence of NMO stretch onthe estimated AVO coefficients. The result from a field data example, the MobilAVO dataset, shows a strong Vp/Vs anomaly in the middle of the section thatmay be a potential hydrocarbon indicator.

INTRODUCTION

Conventional AVO analysis extracts the intercept and slope from NMO-correctedCMP gathers. Since no imaging capability is incorporated, diffraction energy is notproperly analyzed. Diffraction-corrupted intercept and slope sections may lead tofalse hydrocarbon indications. The influence of migration/inversion on AVO analysishas been addressed by several authors (Lumley et al., 1995; Mosher et al., 1996; Dongand Keys, 1997). Lumley et al. (1995) use the conventional common-offset prestacktime migration to collapse the diffraction energy. Mosher et al. (1996) also use theprestack time migration technique. In order to improve the lateral resolution andspatial positioning of AVO anomalies, they choose common-angle sections insteadof common-offset sections. The most important problem with the time migrationschemes is that they cannot handle large lateral velocity and structure variations

1email: [email protected],wenjie [email protected]

275

276 Sun & Dong SEP–97

easily. The time imaging error will consequently produce mispositioning of AVOanomalies in the spatial domain. Dong and Keys (1997) propose a prestack depthinversion scheme. They assume that the earth satisfies a locally 1-D layered veloc-ity model to make the algorithm efficient. The inversion is of Kirchhoff-typed andimplemented in the common-midpoint gather. This locally 1-D assumption restrictstheir schemes to handle only moderate lateral velocity and structure variations. Inthis paper, we propose an AVO inversion scheme for 2-D media. The WKBJ Green’sfunction is calculated by a finite-difference algorithm. A 2.5-D Kirchhoff integral isused in the inversion. Common-image gathers (CIG) are produced as a by-product,that can be used to quality-control the accuracy of velocity model. We first derivea new form of the 2.5-D Kirchhoff integral formula in V (x, z) media and relate it tothe WKBJ Green’s function. Then we discuss the characteristics of the weightingfunction in the Kirchhoff integral. We show the effect of the integral aperture on theestimated amplitude of the reflection coefficient. Finally, we discuss the results ofapplying the new algorithm to synthetic and field data.

THEORY OF 2.5-D KIRCHHOFF INVERSION IN V (X,Z) MEDIA

In the spatial and frequency domain, the 3-D acoustic wave equation can be formu-lated as [

∇2 +ω2

c2(x)

]G(x,xs, ω) = −δ(x− xs) (1)

where G(x,xs, ω) can be approximated by the WKBJ Green’s function

G(x,xs, ω) ∼ A(x,xs)eiωτ(x,xs) (2)

where τ(x,xs) is the traveltime from source xs to an arbitrary point x. Using theWKBJ Green’s function, Beylkin (1985) gave an inversion formula in 3-D media

α(x) ∼ c2(x)

8π3

∫ ∫

S0

d2ξ|h(x, ξ)|

A(x,xs)A(x,xr)

∫dωF (ω)e−iω[τ(x,xs)+τ(x,xr)]D(ω, ξ). (3)

In the above formula, α(x) is the perturbation to the background velocity c(x). Theupdated velocity model is given by

v−2(x) = c−2(x) [1 + α(x)] . (4)

S0 is the 2-D integral surface. h(x, ξ) is introduced by Beylkin (1985), which is asso-ciated with the ray curvature. A(x,xs) and A(x,xr) are the WKBJ Green’s function.F (ω) is a high-pass filter determined the source. D(ω, ξ) represents the observeddata at xr due to the source xs. Bleistein et al.(1987) specialize the 3-D formula tothe 2.5-D geometry using the method of stationary phase. The corresponding 2.5-Dinversion formula is

α(x) ∼ 2

√2

π

∫dξ[1 + c2(x)ps · pr

]

SEP–97 AVO inversion 277

√1

σs0+

1

σr0

(ns · ps0

AsArσs0 + nr · pr0

ArAsσr0

)

∫ dω√iωF (ω)e−iω(τs+τr)D(ω, ξ) (5)

Here, ps and pr are the slowness vectors at the imaging location pointing to the sourceand receiver respectively. σs0 and σr0 are the parameters defined by the followingequations

σs0 =∫ τs

0c(x)dτ, σr0 =

∫ τr

0c(x)dτ. (6)

ns and nr are unit downward normals at the source and receiver points respectively.ps0 and pr0 are the slowness vectors at the source and receiver points respectively.This inversion formula is only valid in the high-frequency limit. Under such circum-stances, it is better to process data for the upward normal derivative ∂α/∂n at eachdiscontinuity surface of α(x). ∂α/∂n is a sum of weighted singular functions withpeaks on the reflectors. Therefore, ∂α/∂n actually provides an image of the sub-surface. Using the Fourier transform, we can obtain the following 2.5-D formula for∂α/∂n.

∂α

∂n(x) ∼ 4√

πc(x)


] 32

[AsAr]−1

√1

σs0+

1

σr0

[ns · ps0A2

s(x,xs)σs0 + nr · pr0A2r(x,xr)σr0

]

∫dω√iωF (ω)e−iω(τs+τr)D(ω, ξ) (7)

Bleistein et al.(1987) also shows that ∂α/∂n can be related to the reflection coefficienton the interface by

∂α

∂n∼ 4 cos2 θR(x, θ)γ(x) (8)

γ(x) in the singular function of the model space. R(x, θ) is determined by the changesof velocity and density above and below the interface and the incident angle on theinterface

R(x, θ) =cdown(x)ρdown cos θ − ρup

√c2up(x)− c2

down(x) sin2 θ

cdown(x)ρdown cos θ + ρup√c2up(x)− c2

down(x) sin2 θ. (9)

In order to determine R(x, θ) from ∂α/∂n, we have to determine cos θ. In their paper,Bleistein et al.(1987) proposed another inversion operator β(x, z) with a kernel slightlymodified from that in ∂α/∂n.

β(x) ∼ 2√πc(x)


] 12

[AsAr]−1

√1

σs0+

1

σr0

[ns · ps0A2


]



There is a simple relation between ∂α/∂n, β(x, z), and cos θ, that is

cos2 θ =∂α∂n

(peak)

4β(peak). (11)

With cos θ known, we can use ∂α/∂n and cos θ to calculate the reflection coefficientR(x, θ). From R(x, θ) and cos θ, we can further estimate the AVO coefficients: in-tercept and slope. Instead of using ∂α/∂n and β(x, z), we propose another pair ofinversion operators that can determine cos θ in a similar, but more straightforwardand physically meaningful manner. The first operator gives the reflection coefficientat the specular incident angle

R(x, θ) ∼ 1√πc(x)

∫dξ

[AsAr]−1

√1

σs0+

1

σr0

[ns · ps0A2


]


The second gives the reflection coefficient multiplied by cos θ

R′(x, θ) ∼ 1√2πc(x)


] 12

[AsAr]−1

√1

σs0+

1

σr0

[ns · ps0A2


]


From R(x, θ) and R′(x, θ), we can easily calculate cos θ

cos θ =R′(x, θ)

R(x, θ). (14)

In order to reduce the sensitivity of cos θ to noise in the data, we use a least-squaresprocedures to estimate cos θ. First, we define a small window (nx × nz). Within thewindow, we can get a series of equations

R(x1, z1, θ) cos θ = R′(x1, z1, θ)R(x2, z2, θ) cos θ = R′(x2, z2, θ)

···

R(xn, zn, θ) cos θ = R′(xn, zn, θ)

(15)

the least-squares sense estimate of cos θ is then

cos θ =

∑nxi,zi

R(x1, z1, θ)R′(x1, z1, θ)∑n

xi,ziR2(x1, z1, θ)

. (16)


AVO THEORY IN ACOUSTIC AND ELASTIC MEDIA

Under the assumption of small incident angle, there is a well-known linearized Zoep-pritz equation (Aki and Richards, 1980). Because we only consider the incident angleless than 35 degree, we have omitted the C term in the original form. For the acousticand elastic media, the expressions for the reflection coefficients are different.

Acoustic AVO approximation

R ≈ A+B tan2 θ (17)

whereA ≈ 1

2

(δVV

+ δρρ

)

B ≈ 12δVV

(18)

Elastic AVO approximation

R ≈ A +B sin2 θ (19)

whereA ≈ 1

2

(δVpVp

+ δρρ

)

B ≈ 12δVpVp− 2

(VsVp

)2 (2 δVsVs

+ δρρ

) (20)

Using the reflection coefficient R and specular incident angle θ, we find the solutionfor intercept and slope is a least-squares problem.

R1 = A+Bf(θ1)R2 = A+Bf(θ2)

···

Rn = A+Bf(θn)

(21)

The resulting estimates of A and B are given by

[AB

]=

[N

∑Ni f(θi)∑N

i f(θi)∑Ni f

2(θi)

] [ ∑Ni Ri∑N

i Rif(θi)

](22)

Getting AVO intercept and slope is not our final goal. The purpose of AVO analysisis to display the Vp/Vs anomaly in the subsurface. This anomaly is a very importanthydrocarbon indication, especially for gas-charged reservoirs. Here we use the fluid-line technique to highlight this anomaly. Assume there is a linear relation

AX +B = 0 (23)


between intercept A and slope B. We specify a window with reasonable size anduse least squares algorithm to estimate the coefficient X. Similar to cos θ, we get anexpression for X

X =

∑nxi,zi

A(xi, zi)B(xi, zi)∑nxi,zi

A2(xi, zi). (24)

The AX+B section is called the fluid-line section, which highlights the Vp/Vs anomaly.

PARAMETER ANALYSIS

The 2.5-D Kirchhoff inversion can be viewed as a weighted Kirchhoff depth migration.In other words, if there is no middle row in equation (12), the final result will be a2-D Kirchhoff depth migration in V (x, z) media. In this section, we investigate therelationship between the two key components in equation (12) in the homogeneousmedium.

Weighting function

The weighting function determines the contribution of each data sample to the image.The weighting function depends on the locations of source, receiver, and diffractor.

w(xs,xr; x) = [AsAr]−1

√1

σs0+

1

σr0

[ns · ps0A2


]

(25)

Double-square-root (DSR) equation

The DSR equation is the kinematic relation between source, receiver, and diffractorin the homogeneous media.

τ(xs,xr; x) = τs + τr

=1

V

(√(xs − x)2 + (zs − z)2 +

√(xr − x)2 + (zr − z)2

)(26)

It is worth investigating the relationship between these two components and otherparameters, such as image depth, integral aperture, velocity, and offset, etc. In orderto simplify the problem, we assume a homogeneous media and discuss the dependenceof weighting function and DSR equation on other parameters, such as offset, depth,and velocity. The DSR equation is a function of imaging depth, velocity, and offset.As shown in Figure 1, with increasing imaging depth, the hyperbolic curve becomesflatter. Therefore, anti-aliasing requirements in the deep zone are not as severe as itis in the shallow zone. Similarly, high velocity corresponds to a flattened hyperbola.Large offset has a similar effect. Actually, if we view the offset response in 3-D, it isthe famous ”Cheops pyramid” (Claerbout, 1985). We then take the first and second


derivatives of the hyperbolic curves. As show in Figure 2, with the increase of offset,the first derivative has two inflection points. Correspondingly, two peak values showup in the second derivative for non-zero offset. In a constant velocity medium, the

-1.0 -0.5 0 0.5 1.0x10 4Midpoint (m)

2

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e (

se

c)

Velocity

-1.0 -0.5 0 0.5 1.0x10 4Midpoint (m)

4

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12

Tim

e (

se

c)

Depth

-1.0 -0.5 0 0.5 1.0x10 4Midpoint (m)

2

4

6

8

10

Tim

e (

se

c)

Offset

Figure 1: Cheops pyramid changes with different parameters. (L) From top to bot-tom, the hyperbolic moveout curves become flatter when velocity increases. (M) Frombottom to top, the curves become flatter with the increase of depth. (R) From bot-tom to top, the hyperbolic curves change from zero to nonzero offset. yalei1-cheops[NR]

weighting function depends only on imaging depth and offset. As shown in Figure 3,the weighting function has a double-peaked shape in non-zero offset. This feature isvery interesting. Intuitively, it is very natural to think that the data value locatedright in the middle of the panel should have the largest contribution to the image. Thedouble-peaked weighing function in the case of common-offset configuration suggeststhat the largest contribution to the image is not from the middle of the integral curve,but from the two flanks. Therefore, it is very important to include the two peaks toget a true-amplitude image when choosing the integral aperture.

SYNTHETIC DATASET

Figure 4 shows a simple 2-D acoustic model based on one used by Dong and Keys(1997), as shown in Figure 4. The only difference is that all the layers here have a10 degree dipping angle. For the first interface, there is no velocity change and onlydensity change. According to equation (9), the reflection coefficient is constant, 0.05.Similarly, we can reach the same result from the acoustic AVO approximation. Thesecond layer has changes in velocity and density, but in opposite signs. Therefore,these two changes cancel each other out and give a zero-valued intercept. Slope B


-1.0 -0.5 0 0.5 1.0x10 4Midpoint (m)

2

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Tim

e (

se

c)

Cheops pyramid

-1.0 -0.5 0 0.5 1.0x10 4Midpoint (m)

-1.0

-0.5

0

0.5

1.0

x10 -3

Tim

e (

se

c)

First derivative

-1.0 -0.5 0 0.5 1.0x10 4Midpoint (m)

1

2

3

4

5

6

x10 -7

Tim

e (

se

c)

Second derivative

Figure 2: Cheops pyramid, first, and second derivatives. (L) Cheops pyramid changesfrom zero to nonzero offset. (M) The first derivative has two inflections points nearthe middle of the panel in the case of non-zero offset. (R) The second derivative hasa corresponding double-peaked shape in non-zero offset. yalei1-deri [NR]

-1.0 -0.5 0 0.5 1.0x10 4Midpoint (m)

60

80

100

120

140

160

Ampl

itude

Depth

-1.0 -0.5 0 0.5 1.0x10 4Midpoint (m)

60

80

100

120

140

Ampl

itude

Offset

Figure 3: The weighting function changes with depth and offset. (L) With increasingdepth, the double peaks smear out. (R) From zero to nonzero offset, the weightingfunction goes from single-peaked to double-peaked shape. yalei1-weight [NR]


is equal to 0.05. Reflection coefficient R increases from zero to nonzero value withthe increase of the incident angle. The third interface has only a velocity change andno density change. The velocity drops across the interface and results in a negativeintercept and slope. We use an acoustic modeling program developed by Dong, which

Depth

(m)

Midpoint (m)

ρ

ρ

ρ

ρ

A=0.05 B=0.0

A=0.0 B=0.05

A=−0.05 B=−0.05

c=1737.4 (m/s)

c=1920.2 (m/s)

2000

3000

4100

=2.1 (g/cm^3)

=1.9 (g/cm^3)

c=1737.4 (m/s)

=1.9 (g/cm^3)

c=1737.4 (m/s)=1.9 (g/cm^3)

100

Figure 4: Dipping acoustic velocity model used in generating the synthetic dataset.yalei1-model [NR]

is based on the reflectivity method (Muller, 1985). For such kind of layer model, themodeling result is not only kinematically, but also dynamically exact. As shown inthe following result, such an accurate modeling program is very helpful for verifyingthe performance of our inversion program. Figure 5 is a common-shot gather. Thefirst two events have a similar pattern, except that the second one goes to a zero-valued amplitude in the near offset. However, the third event shows an oppositepattern. Figure 6 shows an image gather from the inversion result. Since the correctvelocity model has been used in calculating the WKBJ Green’s function, the threeevents have been flattened in the image gather. However, due to the NMO stretchingeffect, the events broaden from near to far offset. One way to check the accuracy ofour inversion result is to pick the peak amplitude along the three events and thencompare it with the theoretical solution. Figure 7 shows that the numerical resultsmatch the theoretical ones very accurately. Figure 8 and 9 shows the intercept A andslope B estimated from the inversion result. Compared with the theoretical resultsunder acoustic approximation, our solution matches the theoretical one very well.These two figures also show the stretching effect very clearly. How to remove thisstretch effect efficiently is our next research topic.


0

1

2

3

4

Time (s

ec)0 1000 2000 3000 4000

Offset (m)

Common-Shot Gather

Figure 5: Common-shot gather generated from the dipping velocity model using thereflectivity method. yalei1-shot [NR]

1000

1500

2000

2500

3000

3500

4000

Dept

h (m

)

0 1000 2000 3000 4000Offset (m)

Smooth Model (R)

1000

1500

2000

2500

3000

3500

4000

Dept

h (m

)

0 1000 2000 3000 4000Offset (m)

Smooth Model (Rcos)

Figure 6: Common-image gather of the inversion result. (L) R as a function of offset.(R) R cos θ as a function of offset. yalei1-dip-cig [NR]


0 1000 2000 3000 4000Offset (m)

0

0.02

0.04

0.06

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Ma

gn

itud

eCoeff R (smooth)

0 1000 2000 3000 4000Offset (m)

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Coeff R (theory)

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0 1000 2000 3000 4000Offset (m)

0

0.02

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0.06

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gn

itud

e

Coeff Rcos (theory)

Figure 7: Comparison of numerical result and theoretical result. (TL) Numeri-cal R. (TR) Theoretical R. (BL) Numerical R cos θ. (BR) Theoretical R cos θ.yalei1-compare [NR]


Figure 8: AVO coefficients A and B. (Top) Intercept A. (Bottom) Slope B. Thestretch effect is very obvious in the first wavelet of slope B. Since the transmissioneffect has not been taken into account, the absolute values for the second and thirdevents are less accurate. yalei1-dip-avo [NR]


Figure 9: Crossplot of intercept A and slope B. The solid curve represents the firstevent, the dashed-line curve corresponds to the second one, and the dashed-dottedcurve is linked with the third event. The swirly nature of the curves is due to NMOstretch (Dong, 1996). The extent of stretching effect can be evaluated by the distancebetween the curves and the original point. yalei1-crossplot [NR]


THE MOBIL AVO DATA AND ITS RESULT

The Mobil AVO dataset is a marine dataset collected from the North Sea. The datasetcontains strong water-bottom multiples. Before AVO analysis, some processing pro-cedures have been applied to the data to remove the multiple energy. All together,there are 952 CMP gathers with 25m sampling intervals. Each gather contains 60traces, the offset sampling interval is 50m and the near offset is 288m. The tracelength is 1000 samples (sampling rate = 4ms). There are two well logs available atCMP-809 and CMP-1571. In well CMP-809, the density, Vp, and Vs were recordedfrom 1km to 3.15km. On the basis of this well’s information, Dong and Keys (1997)built up a 12-layer (some with vertical gradient) background velocity model for theinversion. Since our new approach can output common-image gathers (CIG), we ini-tially use this model in our inversion and then check the accuracy of this layeredmodel. As shown in 10, the events from the old velocity model bend upwards, whichmeans the interval velocity in the old model is lower than the correct one. We thenconducted a conventional velocity analysis. After converting the RMS velocity modelinto an interval velocity model, we applied the new velocity model to the dataset andproduced the new common-image gather at the same CMP location. It is clear thatthe new velocity model is better for imaging and inversion (Figure 10). By stackingthe common-offset inversion result, we got a R and R′ section (Figure 11). After ob-taining the Kirchhoff inversion result, we estimated the cosine of the specular angle θ.According to the elastic AVO approximation theory, we estimated of intercept A andslope B, as shown in Figure 12. Furthermore, we combined the intercept and slopesections and produced a fluid-line section, which shows the Vp/Vs anomaly (Figure13).

CONCLUSION AND DISCUSSION

We implemented an AVO inversion algorithm in V (x, z) media. Our approach is atwo-step inversion scheme:

• 2.5-D Kirchhoff inversion;

• AVO coefficient estimation.

Since the velocity model used in AVO analysis is relatively smooth, the finite-differenceforward modeling result is accurate enough in Kirchhoff inversion. Both the syntheticand field data example can verify the accuracy of the finite-difference scheme. Onthe basis of Bleistein et al. (1987), we proposed another pair of Kirchhoff inversionoperators that have a more obvious physical meaning. One is the specular reflectioncoefficient R, and the other is R multiplied by the cosine of half of the specular inci-dent angle, R′ = R cos θ. The reflection coefficient R, organized into common-imagegathers, is not only necessary in estimating the AVO intercept A and slope B, but also


Figure 10: Common-image gathers of the Mobil AVO dataset. (L) CIG from theold velocity model. (R) CIG from the new velocity model. Both are from the samemidpoint location. The new model is significantly better than the old one. In the oldCIG, because of the use of a low velocity model, not only can the image event not beflattened, it also has a depth shift from top to bottom. yalei1-mobil-cig [NR]


Figure 11: The stacked section of the inversion result. (Top) Reflection coef-ficient R. (Bottom) R cos θ. The diffraction energy has been well collapsed.yalei1-mobil-stack [NR]


Figure 12: AVO coefficient sections of the Mobil AVO dataset. (Top) Intercept Asection. (Bottom) Slope B section. Similar to the stacked section, the diffractionenergy has also been well collapsed in the A and B section. Generally, intercept Aand slope B have opposite polarities. Slope B has a larger value than intercept A.yalei1-mobil-avo [NR]


Figure 13: The fluid-line section of the Mobil AVO dataset. Many strong events inA & B section have been canceled. The strong event in the middle of the sectionshows the anomaly of Vp/Vs, which may be an indicator of hydrocarbon in that area.

yalei1-fluid-line [NR]


essential to update the velocity model. Through checking common-image gathers, wecan update the velocity model and produce a more accurate image. This feature willalso prevent the velocity error from propagating into the final AVO coefficients. Oneof the fundamental differences between Kirchhoff inversion and Kirchhoff depth mi-gration is that Kirchhoff inversion has an extra weighting function varying along theintegral curves. We investigated the relationship between the weighting function anddouble-square-root (DSR) equation in the homogeneous medium. It is interesting tosee that the weighting function has double peaks in the common-offset configuration.This observation tells us that the largest contribution to the image is not from themiddle of the integral curve, but from the two flanks. Therefore, it is very impor-tant to include the locations of the two peaks in order to recover a true-amplitudeimage. We applied our algorithm to both synthetic and field datasets. The syntheticexample shows that this new scheme is very accurate in calculating the reflection co-efficient and the specular incident angle. When applying our approach to the MobilAVO dataset, we updated the velocity model according to the common-image gath-ers. Furthermore, we estimated the AVO coefficients, intercept A and slope B, andthen created a fluid-line expression of Vp/Vs anomaly. Our result shows that there isa strong Vp/Vs anomaly in the middle section that suggests a potential hydrocarbonindicator.

REFERENCES

Aki, K., and Richards, P. G., 1980, Quantitative seismology: Theory and methods:W. H. Freeman and Co., New York.

Beylkin, G., 1985, Imaging of discontinuities in the inverse scattering problem byinversion of a casual generalized Radon transform: J. Math. Phys., 26, 99–108.

Bleistein, N., Cohen, J. K., and Hagin, F. G., 1987, Two and one-half dimensionalBorn inversion with an arbitrary reference: Geophysics, 52, no. 1, 26–36.

Claerbout, J. F., 1985, Imaging the Earth’s Interior: Blackwell Scientific Publications.

Dong, W., and Keys, R. G., 1997, Multi-parameter seismic inversion for hydracarbondetection: 15th World Petroleum Congress Proceedings.

Dong, W., 1996, Fluid-line distortion due to migration stretch: 66th Ann. Internat.Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1345–1348.

Lumley, D. E., Nichols, D., Ecker, C., Rekdal, T., and Berlioux, A., 1995, Amplitude-preserved processing and analysis of the Mobil AVO data set: SEP–84, 125–152.

Mosher, C. C., Keho, T. H., Weglein, A. B., and Foster, D. J., 1996, The impact ofmigration on AVO: Geophysics, 61, no. 6, 1603–1615.

Muller, G., 1985, The reflectivity method: a tutorial: J. Geophys., 58, 153–174.

avo inversion in v x;z media - stanford university

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